From Space to Spacetime

Since antiquity there has been a fascination with the notions of space and time with Aristotle’s philosophy remaining dominant until the advent of the heliocentric Copernican system of the Solar System marked the first steps of modern rational science in its understanding of these concepts. This culminated in the Newtonian theory of familiar three-dimensional space and absolute time. However, the absence of a supposed ether, as established by Michelson and Morley, ushered in the Special Theory of Relativity and the entwined relationship between space and time, whilst Einstein’s General Theory revealed a more complicated geometry of the two through curved spacetime. This conference traced our understanding of space and time across the ages up to the latest knowledge of spacetime and the expanding Universe.

Hesiod was an ancient Greek poet, active between 750 and 650 BC, who wrote a poem describing the origins and genealogies of the Greek gods.

First of all there was Chaos (a gaping void) in the universe and from this Chaos came everything. After Chaos came Gaia (Earth), Tartarus (the cave-like space under the earth; the later-born Erebus is the darkness in this space), and Eros (representing sexual desire – the urge to reproduce – instead of the emotion of love as is the common misconception). Hesiod made an abstraction because his original chaos is something completely indefinite.

Eventually Zeus became the king of the Gods and created a secure dwelling place for the immortals forever, the peak of snowy Olympus and the Universe as we know it was complete.

Anaximander (c. 610 – c. 546 BC) was a pre-Socratic Greek philosopher who lived in Miletus, a city of Ionia (in modern-day Turkey).

Little of his life and work is known today. According to available historical documents, he is the first philosopher known to have written down his studies, although only one fragment of his work remains. Fragmentary testimonies found in documents after his death provide a portrait of the man. He introduced the first abstract nature of philosophy.

He was an early proponent of science and tried to observe and explain different aspects of the universe, with a particular interest in its origins, claiming that nature is ruled by laws, just like human societies, and anything that disturbs the balance of nature does not last long.

In physics, his postulation that the indefinite (or apeiron) was the source of all things led Greek philosophy to a new level of conceptual abstraction. He is described as the precursor of modern physics.

Parmenides of Elea (late sixth or early fifth century BC) was a pre-Socratic Greek philosopher from Elea in Magna Graecia (Greater Greece, included Southern Italy). He was the founder of the Eleatic school of philosophy.

The single known work of Parmenides is a poem, On Nature, which has survived only in fragmentary form. In this poem, Parmenides prescribes two views of reality. In “the way of truth” (a part of the poem), he explains how reality (coined as “what is-is”) is one, change is impossible, and existence is timeless, uniform, necessary, and unchanging. It is generally considered to be one of the first digressions into the philosophical concept of being.

He made an argument against nothingness, essentially denying the possible existence of a void. He is considered to be the first metaphysician.

Zeno of Elea (c. 490 – c. 430 BC) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides.

Although many ancient writers refer to the writings of Zeno, none of his works survive intact. The main sources on the nature of Zeno’s arguments on motion, in fact, come from the writings of Aristotle and Simplicius of Cilicia. His arguments are perhaps the first examples of a method of proof called reductio ad absurdum, literally meaning to reduce to the absurd. He is also regarded as the first philosopher who dealt with the earliest attestable accounts of mathematical infinity.

Zeno’s paradoxes are a set of philosophical problems to support Parmenides’ doctrine that contrary to the evidence of one’s senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. A paradox is defined as a “seemingly absurd or self-contradictory statement that when investigated may prove to be well founded or false” [Oxford Dictionary].

Some mathematicians and historians, such as Carl Boyer, hold that Zeno’s paradoxes are simply mathematical problems, for which modern calculus provides a mathematical solution. Some philosophers, however, say that Zeno’s paradoxes and their variations remain relevant metaphysical problems.

Zeno argued that any quantity of space (or time) must either be composed of ultimate indivisible units or it must be divisible ad infinitum. If it is composed of indivisible units, then these must have magnitude and we are faced with the contradiction of a magnitude which cannot be divided. If, however, it is divisible ad infinitum, then we are faced with the different contradiction of supposing that an infinite number of parts can be added up to make a merely finite sum.

Toward the end of the introduction to his analysis of place, Aristotle notes that “Zeno’s difficulty requires some explanation; for if everything that is is in a place, it is clear that there will also be a place of the place, and so on to infinity” (Arist. Ph. 4.1, 209a23–5). His subsequent statement of the problem is even briefer but adds one key point: “Zeno raises the problem that, if place is something, it will be in something” (Arist. Ph. 4.3, 210b22–3; cf. Eudemus fr. 78 Wehrli, [Arist.] De Melisso Xenophane Gorgia 979b23–7, Simp. in Ph. 562, 3–6). Zeno would appear to have argued as follows. Everything that is is in something, namely a place. If a place is something, then it too must be in something, namely some further place. If this second place is something, it must be in yet another place; and the same reasoning applies to this and each successive place ad infinitum. Thus, if there is such a thing as place, there must be limitless places everywhere, which is absurd. Therefore, there is no such thing as place. This argument could well have formed part of a more elaborate argument against the view that there are many things, such as that if there are many things, they must be somewhere, i.e. in some place; but there is no such thing as place and thus no place for the many to be; therefore, there are not many things. This is, however, only speculation.

The atomistic void hypothesis was a response to the paradoxes of Parmenides and Zeno, the founders of metaphysical logic, who put forth difficult to answer arguments in favour of the idea that there can be no movement. They held that any movement would require a void—which is nothing—but a nothing cannot exist. The Parmenidean position was “You say there is a void; therefore the void is not nothing; therefore there is not the void”. The position of Parmenides appeared validated by the observation that where there seems to be nothing there is air, and indeed even where there is not matter there is something, for instance light waves.

The atomists agreed that motion required a void, but simply ignored the argument of Parmenides on the grounds that motion was an observable fact. Therefore, they asserted, there must be a void. This idea survived in a refined version as Newton’s theory of absolute space, which met the logical requirements of attributing reality to not-being. Einstein’s theory of relativity provided a new answer to Parmenides and Zeno, with the insight that space by itself is relative and cannot be separated from time as part of a generally curved space-time manifold. Consequently, Newton’s refinement is now considered superfluous.

Leucippus (5th cent. BCE) is reported in some ancient sources to have been a philosopher who was the earliest Greek to develop the theory of atomism—the idea that everything is composed entirely of various imperishable, indivisible elements called atoms.

Leucippus and Democritus proposed the following model in the fifth century B.C.

1. Matter is composed of atoms separated by empty space through which the atoms move.

2. Atoms are solid, homogeneous, indivisible, and unchangeable.

3. All apparent changes in matter result from changes in the groupings of atoms.

4. There are different kinds of atoms that differ in size and shape.

5. The properties of matter reflect the properties of the atoms the matter contains.

Aristotle did not like these. He vehemently opposed the Atomic theory and believed that instead of matter being made of tiny particles (atoms) they were all fundamentally air, fire, water, and earth.

Plato (428/427 or 424/423 – 348/347 BC) was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world. He is widely considered the pivotal figure in the development of Western philosophy. Unlike nearly all of his philosophical contemporaries, Plato’s entire work is believed to have survived intact for over 2,400 years.

Plato was also one of ancient Greece’s most important patrons of mathematics. Inspired by Pythagoras, he founded his Academy in Athens in 387 BCE, where he stressed mathematics as a way of understanding more about reality. In particular, he was convinced that geometry was the key to unlocking the secrets of the universe. The sign above the Academy entrance read: “Let no-one ignorant of geometry enter here”.

Timaeus is one of Plato’s dialogues, mostly in the form of a long monologue given by the title character Timaeus of Locri, written c. 360 BC. The work puts forward speculation on the nature of the physical world and human beings and is followed by the dialogue Critias. He presents an elaborately wrought account of the formation of the universe and an explanation of its impressive order and beauty. The universe, he proposes, is the product of rational, purposive, and beneficent agency. It is the handiwork of a divine Craftsman (“Demiurge,” dêmiourgos, 28a6) who, imitating an unchanging and eternal model, imposes mathematical order on a preexistent chaos to generate the ordered universe (kosmos).

Armillary Sphere is a model of objects in the sky (on the celestial sphere), consisting of a spherical framework of rings, centred on Earth or the Sun, that represent lines of celestial longitude and latitude and other astronomically important features, such as the ecliptic. As such, it differs from a celestial globe, which is a smooth sphere whose principal purpose is to map the constellations. It was invented separately in ancient Greece and ancient China, with later use in the Islamic world and Medieval Europe. The Greeks considered it to show the shape of the world’s soul. The geometry of space depends on the combination of forms chosen by an ideal thinker (the Demiurge)

Pythagoreans (in particular Philolaus and Archytas) of ancient Greece were the first researchers known to have investigated the expression of musical scales in terms of numerical ratios, particularly the ratios of small integers. Their central doctrine was that “all nature consists of harmony arising out of numbers”. From the time of Plato, harmony was considered a fundamental branch of physics, now known as musical acoustics.

In order to trace the origins of the concept of space in Greek antiquity, we should first define our own the concept of space. Our problem for defining space for this purpose is that we have modern prejudices concerning our own modern concept of space and we must look back in history to a time when the concept of space, as we have come to know it, was at the most rudimentary and barely recognizable if it existed at all. The simplest and most useful definition of space for tracing the development has been given by Smart. “When we naively begin to think about space we most naturally think of it as though it were either some all-pervading stuff or some sort of receptacle.” (Smart, p.5). This seems to be approximately the point where the Greeks began their inquiry and is quite different from our own present concept of space, but serves us well if we consider that the evolution of the concept can be considered as a constant defining and refining of the terms and properties which we have come to use to characterize space.

The Greek concept of space began a new period of scientific maturity only when Plato asked ‘why do we need space?’ At that point in time, the concept of space became necessary for the further development of a physics of reality. The special recognition that space existed as an independent physical ‘thing’ was a necessary prerequisite to study and understand motion and thus played an important role in man’s view of the universe about him.

The concept of space was needed to provide a constant background by which motion and change could be defined. Until the ancient Greeks were able to develop a concrete notion of space, many problems arose concerning motion, change, Being, Non-Being, the discrete, continuity and so on. These qualities of physical reality can only be defined and understood against some constant physical reality which we call space. It would be ludicrous to think that the Greeks solved all of the problems concerning the concept of space, since even today scientists and philosophers are not sure whether space is discrete or continuous in reality. However, it is to the credit of the Ancient Greek philosophers and their civilization that they were able to define space as well as they did.

The function of the Receptacle is to receive all things in it. It never takes on any character of the things that enters it. It is moulded and remoulded into various shapes. It is a kind of “matrix” (moulding-stuff) that appears to have different qualities at different times, nonetheless, always stays the same. For instance you can make a figure out of gold, but you can change that figure and make it a different shape. Therefore the only thing you can say about the different figures made is that they are made of gold. You could never describe the figures as “being” considering they can change. Rather you should be content to give them a title of “suchlike”

Plato believed that he could describe the Universe using five simple shapes. These shapes, called the Platonic solids, did not originate with Plato. In fact, they go back thousands of years before Plato.

Platonic solids are any of the five geometric solids whose faces are all identical, regular polygons meeting at the same three-dimensional angles. Also known as the five regular polyhedra, they consist of the tetrahedron (or pyramid), cube, octahedron, dodecahedron, and icosahedron.

There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one’s hand when picked up, as if it is made of tiny little balls. By contrast, a highly nonspherical solid, the hexahedron (cube) represents “earth”. These clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cube’s being the only regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth.

Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks, “…the god used [it] for arranging the constellations on the whole heaven”. Aristotle added a fifth element, aithēr (aether in Latin, “ether” in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato’s fifth solid.

The Receptacle can take one of these shapes in the physical cosmos.

Aristotle is considered to be like modern analytical philosophers (although he did get things wrong). He was a critic of Plato’s ideas about the characteristics of “place”.

Characteristics of place which need to be accounted for:

1) Place is what contains that of which it is the place

2) Place is no part of the thing

3) The immediate place of a thing is neither less nor greater than the thing

4) Place can be left behind by the thing and is separable. In addition:

5) All place admits of the distinction of up and down, and each of the bodies is naturally carried to its appropriate place and rests there, and this makes the place either up or down.

Place, as an inner surface of a volume, is also immobile.

…Just, in fact, as a vessel is a transportable place, so place is a non-portable vessel

The Aristotelian Cosmos contains 4 elements: air; fire; water, earth. These were acted on by two forces, gravity (the tendency of earth and water to sink) and levity (the tendency of air and fire to rise). He later added a fifth element, aether, to describe the void that fills the universe above the terrestrial sphere.

Proclus of Lycia (8 February 412 – 17 April 485 AD) was a Greek Neoplatonist philosopher, one of the last major classical philosophers. He set forth one of the most elaborate and fully developed systems of Neoplatonism. He stands near the end of the classical development of philosophy, and was very influential on Western medieval philosophy (Greek and Latin).

The majority of his works are commentaries on dialogues of Plato (Alcibiades, Cratylus, Parmenides, Republic, Timaeus). In these commentaries he presents his own philosophical system as a faithful interpretation of Plato, and in this he did not differ from other Neoplatonists, as he considered the Platonic texts to be divinely inspired (ὁ θεῖος Πλάτων ho theios Platon—the divine Plato, inspired by the gods) and therefore that they spoke often of things under a veil, hiding the truth from the philosophically uninitiated. Proclus was however a close reader of Plato, and quite often makes very astute points about his Platonic sources. A number of his Platonic commentaries are lost.

In addition to his commentaries, Proclus wrote two major systematic works. The Elements of Theology (Στοιχείωσις θεολογική) consists of 211 propositions, each followed by a proof, beginning from the existence of the One (divine Unity) and ending with the descent of individual souls into the material world. The Platonic Theology (Περὶ τῆς κατὰ Πλάτωνα θεολογίας) is a systematisation of material from Platonic dialogues, showing from them the characteristics of the divine orders, the part of the universe which is closest to the One.

We also have three essays, extant only in Latin translation: Ten doubts concerning providence (De decem dubitationibus circa providentiam); On providence and fate (De providentia et fato); On the existence of evils (De malorum subsistentia).

Like the other Neoplatonists he combined Platonic, Aristotelian, and Stoic elements in his thought. He refined and systematised the elaborate metaphysical speculations of Iamblichus. In contrast to the sceptic position that the material universe is outside the human consciousness and can only be known through sensory impressions, the Neoplatonists emphasized the underlying unity of all things and placed the human soul and the material universe in a hierarchy of emanation from a universal being, in which every level is a reflection of that being.

René Descartes (31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and scientist. He asserted that matter is composed of three elements:

1st fire; 2nd air; 3rd Earth

The second element emerged before the other two and was thought to be somehow more fundamental. Together with the first element, they account for all observable changes on the third element by contact-action.

What came to be known as “Cartesian ether” was celestial matter that either consisted of air or air permeated by fire.

Descartes’ conception of the physical world was therefore deeply ethereal since the first two elements explain the effects on the third.

Cartesian plenum: A continuous swirling vortex of matter. Descartes argued that the universe was composed of a “subtle matter” he named “plenum,” which swirled in vortices like whirlpools and actually moved the planets by contact. Here, these vortices carry the planets around the Sun.

A vortex, for Descartes, was a large circling band of material particles. In essence, his vortex theory attempted to explain celestial phenomena, especially the orbits of the planets or the motions of comets, by situating them (usually at rest) in these large circling bands.

Bodies rarefy via the displacement of some matter with other matter.

Boyle and Newton disagreed with Descartes’ view. They believed that empty space could exist.

Sir Isaac Newton PRS FRS (25 December 1642 – 20 March 1726/27) was an English mathematician, astronomer, theologian, author and physicist (described in his own day as a “natural philosopher”) who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution.

Newton’s ideas about ethers were not constant. He constructed several sketchy models. His view was that matter was inert. To explain why nature was animated he had to come up with another theory.

The ultimate answer for him was divine agency. But he was also inclined to find intermediate grounds for it in nature. The solution was to propose that principles like gravity were active. Their source, he speculated, was one or more ethers.

His main account “Hypothesis concerning the properties of light” attributes ether to these properties:

Subtle; Ubiquitous (but not empty space); Particulate; Penetrating (can be found inside bodies); Active (he thought matter was meant to be inert); Sensorially imperceptible but still detectable; A Repulsive force between ether particles (hence elastic); A Repulsive force between them and gross matter particles

There are thus clearly some similarities and differences between this ether and the Cartesian one.

Newton used his ideas about the ether to provide rough explanations of a number of phenomena, e.g. gravitational attraction. The repulsive force between larger ether particles suggested that ether density increased away from the Sun. This meant that the denser ether would push planets towards less dense regions, i.e. towards the Sun. Also the pushing would be greater on the side of the planet furthest from the Sun with the same result.

James Clerk Maxwell FRS FRSE (13 June 1831 – 5 November 1879) was a Scottish scientist in the field of mathematical physics

Initially Maxwell was a firm believer in the ether but he never managed to construct a satisfactory mechanical model.

His 1861/2 molecular vortex model made use of an elastic medium to provide a mechanical account of electromagnetic phenomena.

“The ether was to be conceived as a fluid medium; in a region of nonzero magnetic field, this medium would be filled with innumerable small vortex tubes or filaments, corresponding in geometrical arrangement to the magnetic field lines…” (Siegel 1991; page 60).

Thomas Samuel Kuhn (July 18, 1922 – June 17, 1996) was an American physicist, historian and philosopher of science whose controversial 1962 book The Structure of Scientific Revolutions was influential in both academic and popular circles, introducing the term paradigm shift, which has since become an English-language idiom.

Larry Laudan (born 1941) is a contemporary American philosopher of science and epistemologist.

In particular Laudan argues that history demonstrates the unreliability of success-to-truth inferences.

Success: the ability of a theory to correctly predict

Truth: the overall (approximate) correctness of the theory

Empirical success therefore (approximate) truth so realists are happy (yes) but anti-realists are not happy (No)

He lists several past successful theories that cannot be true since their successors were radically different.

He says “the crystalline spheres of ancient and medieval astronomy; the humoral theory of medicine; the effluvial theory of static electricity; ‘catastrophic’ geology, with its commitment to a universal (Noachian) deluge; the phlogiston theory of chemistry; the caloric theory of heat; the vibratory theory of heat; the vital force theories of physiology; the electromagnetic ether; the optical ether; the theory of circular inertia; theories of spontaneous generation.” (page 33).

The historical record therefore supports the view that our current and future best theories will also turn out to be false.

From radical changes in our conception of the world we can infer that the old conceptions were not close to the truth, but we cannot infer that it had no truth content whatsoever for that would be to equivocate between completely false and partly false (hence partly true).

In reaction to such worries, some realists opt for weaker notions of the truth, the partial truth – see Kuipers (2000).

An example of the divide-et-impera approach is structural realism (SR). Roughly SR holds that we can only know the structure of the unobservable or hidden world. One implication is that the structural parts of an earlier theory are responsible for its success in surviving into a successor theory.

“Poincaré is claiming that … Fresnel entirely misidentified the nature of light, his theory accurately described not just light’s observable effects but its structure. There is no elastic solid ether. There is, however, from the later point of view, a (disembodied) electromagnetic field. The field in no clear sense approximates the ether, but disturbances in it do obey formally similar lawsto those obeyed by elastic disturbances in a mechanical medium. Although Fresnel was quite wrong about what oscillates, he was, from this later point of view, right, not just about the optical phenomena, but right also that these phenomena depend on the oscillations of something or other at right angles to the light.” (page 118)

The conjectural aspect of either arguments, and in particular of the luminiferous/EM ether, was not lost on scientists at the time

Examples:

Faraday – “I do not perceive in any part of space … anything but forces and the lines in which they are exerted … The view which I am so bold to put forth … endeavours to dismiss the ether, but not the vibration.” (1846). In other words get rid of the ether but keep the vibrations.

Michael Faraday FRS (22 September 1791 – 25 August 1867) was a British scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic induction, diamagnetism and electrolysis.

George Airy endorsed the truth of Fresnel’s mathematical theory but remain sceptical about the ether.

1) Realists only identify those parts of theories that are (approximate) true in retrospect

2) Similarly, they only identify those parts of theories that contribute to a theory’s success in retrospect

3) If this is indeed the case, then they trivialise their argument

4) That’s because there is now a guarantee that those parts that survive are both successful and (approximately) true

Can we identify which are the best and worst parts of a theory!

Stanford’s charge must be taken seriously. Some realists have put the cart (survival) before the horse (success). We can in fact identify the parts that contribute towards a theory’s success prospectively – see Votsis (2012).

By logically weakening the premises, we can find out the bare minimum required to derive the correct predictions.

The ether once more:

Support for this view can be found in the case of the ether, if it was not essential in the success of the wave theory. In relation to Fresnel’s theory, “the assumption of an ether … certainly did not occur as a premise in the deduction of those [unsuspected] phenomena [e.g. the Poisson spot], but served rather as a means of reconciling the mathematical theory with the ordinary metaphysical prejudices of the time” (Torretti 2007:360).

We clearly need a sort of commitment conservatism. This does not preclude us from boldly theorising. Yet, even when our theorising is more tame, e.g. when it’s guided by analogy, the results are mixed.

Some analogies work and some fail:

Good – Inverse-square law in electrostatics inspired by gravity; Bad – Vulcan to explain Mercury’s perihelion inspired by Neptune

One can only find out through trying them out, though at some point it may be unproductive to continue doing so.

Summary

1) Even our best theories may be very far away from the truth, e.g. a theory of everything (assuming it exists),

2) Given that no theory is perfectly successful, we should expect some parts of theories to be replaced

3) Successor theories must be such that they degenerate into well-confirmed parts of their predecessors

4) Logically weakening a theory to see if all correct predictions can still be made sifts the good from the bad parts

5) Even in the best cases, when we are guided by analogies in theory construction, the results may be mixed

Meanwhile … though ethers are no longer with us, the positing of subtle, ubiquitous and hard-to-detect entities continues unabated

Answers to questions:

1) Could the Higgs field be ether?

2) Einstein thought there was a connection between relativity and ether

Lecture 3: Dr Vincenzo De Risi (Université Paris-Diderot – CNRS) – From Substance to Function: the Structure of Space in Leibniz and Newton

The 17th century was important and influential and space was an important topic at this time.

The notion of space is peculiarly modern, and its history is a very complicated one. In fact, there were several distinct and irreducible conceptions of space in the early modern age.

These modern notions of space were the result of innumerable debates and developments in the fields of metaphysics, epistemology, geometry, mechanics, geography, astronomy, optics and so forth.

The notion of space was often employed in the renaissance to overturn Aristotelian philosophy. It was largely recognised as a concept that could not fit into the scholastic systems of thought. Telesio, Bruno, Patrizi, etc.

Franciscus Patricius (25 April 1529 – 6 February 1597) was a philosopher and scientist from the Republic of Venice of Croatian descent

Space is not a thing of the world. It is a hypostatical extension, not adhering to anything else. It is not a quantity, but if it were it would not be a quantity of categories.

Patrizi said

“Since space is not bounded by the limits of the bodies, nor by the limits of another space, not by its own limits, nor by the limits of an incorporeal being, we conclude that space is infinite.

If anything, corporeal or incorporeal, is not somewhere, it does not exist.

If God is indivisible, as He is, He will be indivisible in space and circumscribed by space. If He is nowhere, it cannot be though without space. If He is

somewhere, as in heaven or above the heaven, He is surely in space. If He is everywhere, He cannot but be in space.

Therefore all beings, and all things above the beings, are in space.”

Patrizi’s ideas were a great influence in the seventeenth century. In Britain, he was discussed (and his views were often accepted by Bacon, Fludd, Gilbert, Harriot, Werner, Hobbes, Herbert of Cherbury, Barrow and others.

Henry More FRS (12 October 1614 – 1 September 1687) was an English philosopher of the Cambridge Platonist school.

Henry More said

“When we shall have enumerated those names and titles appropriate to space, this infinite immobile, extended (entity) will appear to be not only something real (as we have just pointed out) but even something Divine (which so certainly is found in nature).

Of this kind are the following, which are metaphysicians attribute particularly to the First Being, such as: One, Simple, Immobile, External, Complete, Independent, Existing in Itself, Subsisting by Itself, Incorruptible, Necessary, Immense, Uncreated, Uncircumscribed, Incomprehensible, Omnipresent, Incorporeal, All-penetrating, All-embracing, Being by Its essence, Actual Being, Pure Act

The Very Divine Numen is called by the Cabbalists, MAKOM, that is Place.”

Newton was also influenced by Patrizi. Similar views were found in his conception of space as an infinite three dimensional extension. De gravitatione (1671?)

“Perhaps now it may be expected that I should define extension as substance, or accident, or else nothing at all. But by no means, for it has its own manner of existing which is proper to it and which fits neither substances nor accidents.

Space is an affection of a being just as a being. No being exists or can exist which is not related to space in some way. God is everywhere, created minds are somewhere, and body is in the space that it occupies; and whatever is neither everywhere nor anywhere does not exist. And hence it follows that space is an emanative effect of the first existing being, for if any being whatsoever is posited, space is posited.”

Newton was able to shape these metaphysical views in a useful framework for his new physics. Philosphae naturalis; Principia mathematica (1687, 1713, 1726)

“Absolute space, of its own nature without reference to anything external, always remains homogenous and immovable.

Place is the part of space that a body occupies, and it is, depending on the space, either absolute or relative. I say the part of space, not the position of the body or its outer surface. For the places of equal solids are always equal, while their surfaces are for the most part unequal because of the dissimilarity of shapes; and positions, properly speaking do not have quantity and are not so much places as attributes of places.

God endures always and is present everywhere, and by existing always and everywhere he constitutes duration and space.”

Part 2: Leibniz

Among the proposals concerning new conceptions of space in the early modern age, Leibniz’s notion of space stands out as highly peculiar and unprecedented.

Gottfried Wilhelm (von) Leibniz (1 July 1646 [O.S. 21 June] – 14 November 1716) was a prominent German polymath and philosopher in the history of mathematics and the history of philosophy.

Its sources were rather mathematical than metaphysical and it was especially developed in relation to geometry.

As a young man, Leibniz endorsed some views on space that may be assimilated to those of Newton.

Leibniz in 1668 (to his teacher Jacob Thomasius):

“In fact, space is almost more substantial than the body. Indeed, if you remove a body, space and its extension would remain (which is called a void). While if you remove space, the body cannot remain.

I prove that a figure is a substance, or better that space is a substance and a figure something substantial, from the fact that every science is science of a substance, and geometry is a science.”

Leibniz changed his views during his mathematical studies in Paris (1672-1676), where he learned Cartesian physics, as well as higher mathematics.

“If space were something like an extension, the nature of matter would be to fill space, and motion would be the change of space and therefore something absolute … and all other consequences I have drawn in my theoria motus abstracti. But, on the contrary, space is not a thing, nor motion is anything absolute, but it is a relation.

Space and Time are not things, but rather real relations.”

Leibniz’s final definition of space (1715):

“I hold space to be something purely relative, as time is – that I hold it to be an order of coexistences, as time is an order of successions. For space denotes, in terms of possibility, an order of things that exist at the same time, considered as existing together, without entering into their particular manners of existing. And when many things are seen together, one consciously perceives this order of things among themselves.

I do not say, therefore, that space is an order or situation, but an order of situations, or (an order) according to which situations are disposed, and that abstract space is that order of situations when they are conceived as being possible. Space is therefore something merely ideal.”

The central notion of this relational conception of space is the relation of situs, or situation.

The notion of situs is primitive, and therefore left undefined by Leibniz. He, however, gave a definition by abstraction of situs, defining what it means to have the same situs; two sets of points have the same reciprocal situations if and only if they are congruent (i.e. isometric).

The situs points A, B, C is congruent to or the same of, the situs of points A’, B’, C’.

Space is defined as a system of situational relation.

Given the definition of situation by congruence, space is for Leibniz a system of distances.

It approaches our notion of a metric space.

Extension is not a primitive notion, but it is rather produced by a system of distances.

Since his mathematical studies in Paris, Leibniz had embarked in the project of a new geometry, whose main object was the relation of situation. He called this new geometry an analysis situ.

A central idea in Leibniz’s writings on the analysis situs is to supersede the ordinary ‘geometry of magnitudes’ (classical geometry, recently algebrazied) with a theory that could express spatial notions.

Mathematization of space in the modern tradition. Classical notions of position and place as purely local concepts. Development of a geometry of position.

The geometrical study of the properties of space itself is the most important novelty of Leibniz’s analysis situs, and the crowning feature of the new science of space.

Geometry is no longer a science of figures in space, but a science of space itself as a structure (a situational order).

Leibniz developed theorems (and definitions) about properties like uniformity, homogeneity, dimensionality, continuity and flatness of space.

Even the Parallel Postulate was recognised by him as a property of space rather than as a property of parallel lines (in fact, he grounded it in the notion of similarity).

His attempts in a new geometry were left unpublished, and are now preserved in thousands of pages in the archives of Hanover library.

Part 3: The clash

Leibniz’s and Newton’s conceptions of space were widely discussed in the course of the Leibniz-Clarke correspondence.

The relations between Leibniz and Newton were already very tense because the dispute on the priority of the discovery of calculus had exploded in 1711.

Leibniz about Newton: “Il est mon rival; c’est tout dire.”

In 1714, Duke Georg Ludwig of Hanover, Leibniz’s employer, became King George I of England. His daughter-in-law, Caroline von Ansbach, Leibniz’s pupil and fervent Lutheran, became Princess of Wales.

Leibniz wrote to her in 1715 attacking British philosophy (and Locke) as impious, and especially concentrating on the notion of space – the idolum anglorum.

Newton was preoccupied by Leibniz’s attack and influence at the English court, and suspected that he wanted to further attack him on the priority dispute. He asked his follower Samuel Clarke (1675-1729) to reply to Leibniz’s letter.

Samuel Clarke (11 October 1675 – 17 May 1729) was an English philosopher and Anglican clergyman.

“…elles ne sont pas cerites sans l’avis du chevalier Newton…”

Leibniz attacked Newton’s too metaphysical conception of space:

“If space is an absolute reality, far from being a property of an accident opposed to substance, it will have a greater reality than substances themselves. God cannot destroy it, nor even change it in any respect.

I objected that space, taken for something real and absolute without bodies, would be a thing eternal, unaffected, and independent of God. The author endeavours to elude this difficulty by saying that space is a property of God. In answer to this I have said, in my foregoing paper, that the property of God is immensity but that space (which is often commensurate with bodies) and God’s immensity are not the same thing.

I asked also, if space is a property, what thing will an empty limited space (such as that which my adversary imagines in an exhausted receiver) be the property of? It does not appear reasonable to say that this empty space, either round or square, is a property of God. Will it be then perhaps the property of some immaterial, extended, imaginary substances which the author seems to fancy in the imaginary spaces?

I have still other reasons against this strange imagination that space is a property of God. If it is so, space belongs to the essence of God. But space has parts; therefore there would be parts in the essence of God.”

Newton and Clarke, on the other hand, were never able to understand the notion of a space as a system of relations.

Clarke said:

“Space and time are quantities which situation and order are not.”

Newton said:

“Leibniz saith that space is the order of coexistences and time the order of successive existences: I suppose he means that space is the order of coexistences in space and time the order of successive existences in time, or that space is space in space, and time is time in time.”

In the last letters, the core of the dispute seems to be the relativity of motion, and therefore Newton’s conception of absolute space.

Leibniz said:

“To say that God can cause the whole universe to move forward in a right line or in any other line, without making otherwise any alteration in it, is another chimerical supposition. For two states indiscernible from each other are the same state, and consequently, it is a change without any change. Besides, there is neither rhyme nor reason in it. But God does nothing without reason, and it is impossible that there should be any here. Besides, it would be agendo nihil agere, as I have just now said, because of the indiscernibility. These are idola tribus mere chimeras, and superficial imaginations. All this is only grounded on the supposition that imaginary space is real.”

Clarke said:

“Two places, though exactly alike, are not the same place. Nor is the motion or rest of the universe the same state any more than the motion or rest of a ship is the same state, because a man shut up in a cabin cannot perceive whether the ship sails or not, as long as it moves uniformly.

The motion of the ship, though the man does not perceive it, is a real different state and has real different effects, and, on a sudden stop, it would have other real effects and so likewise would an indiscernible motion of the universe.”

Leibniz said:

“The author replies now that the reality of motion does not depend upon being observed and that a ship may go forward, and yet a man who is in the ship may not perceive it. I answer, motion does not indeed depend upon being observed, but it does depend upon being possible to be observed. There is no mention when there is no change that can be observed. And when there is no change that can be observed, there is no change at all.

Clarke said:

“It is affirmed that the motion of the material universe would produce no change at all, and yet no answer is given to the argument I advanced that a sudden increase or stoppage of the motion of the whole would give a sensible shock to all the parts, and it is as evident that a circular motion of the whole would produce a vis centrifuga in all parts.”

Newton said:

The effects distinguishing absolute motion from relative motion are the forces of receding from the axis of circular motion. For in purely relative circular motion these forces are null, while in true and absolute circular motion they are larger or smaller in proportion to the quantity of motion. If a bucket is hanging from a very long cord and is continually turned around until the cord becomes twisted tight, and if the bucket is thereupon filled with water and is at rest along with the water and then, by some sudden force, is made to turn around in the opposite direction and, as the cord unwinds, perseveres for a while in this motion; then the surface of the water will at first be level, just as it was before the vessel began to move.”

The correspondence ended at this point, and we do not know what Leibniz’s reply to Clarke’s and Newton’s example of the rotating bucket would have been.

The Leibniz-Clarke correspondence was published the following year by Clarke, and underwent several further editions in the 18th century, becoming one of the most read philosophical books in the early modern age.

Newton’s absolute space was generally accepted by most of the physicists in the 18th and 19th centuries, but they seldom endorsed his and Clarke’s metaphysics of space.

Leibniz’s conception of a mathematical space was also discussed at length in the 18th century, and found general acceptance in the 19th century. A new structural approach to geometry.

The Leibniz-Clarke correspondence was further debated in the late 19th century, in connection with Mach’s criticisms to Newton’s absolute space, and again in the years of the first philosophical debates on Einstein’s theory of relativity.

Other scientists were engaged in the work that ended up in Einstein’s General and Special relativity

Introduction:

Euclidean geometry

But at the start of 1907 the use of Euclidean space was challenged as prior to this date space and time were considered separate. Euclidean geometry and separate space and time were no longer considered adequate to the task of describing physical reality. It was replaced by the geometry of a four dimensional space that Minkowski named the “welt” (“world” in English)

Hermann Minkowski (22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number theory, mathematical physics, and the theory of relativity.

Why was it necessary to unite space and time?

“The ether was undetectable and tone could generate the same mathematics without relating to the ether theory”

The papers about special relativity by Einstein in 1905 were based on two postulates

1) The principle of relativity – The laws by which the states of physical systems undergo change are not affected, whether these changes of state referred to the one or the other of two systems in uniform translator motion relative to each other

2) The principle of invariant light speed – “…light is always propagated in empty space with a definite velocity/speed c which is independent of the state of motion of the emitting body” (from the preface). That is light in a vacuum propagates with the speed c (a fixed constant and independent of direction) in at least one system of inertial coordinates (the “stationary system”), regardless of the state of motion of the light source.

After 1905 Minkowski realised that by using welt, where space and time are united one could describe the ideas of Einstein’s equivalence.

The beginning part of his address called “Space and Time” delivered at the 80th assembly of German Natural Scientists and Physicians (21 September 1908) is now famous:

“The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” H. Minkowski

Experimental physicists of the time hated these ideas.

Four months later he gave a talk which made the physicists happier.

He died in 1909 and this changed everything.

Who were the main mathematicians and physicists developing these concepts?

Jakob Johann Laub (born as Jakub Laub, 7 February 1884 in Rzeszów – 22 April 1962 in Fribourg) was a physicist from Austria-Hungary, who is best known for his work with Albert Einstein in the early period of special relativity.

Arnold Johannes Wilhelm Sommerfeld, ForMemRS (5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored a large number of students for the new era of theoretical physics.

Einstein and Laub initially wrote papers against Minkowski but Born and Summerfeld were supportive. Einstein’s opinion may have been coloured by the fact that he was Minkowski’s student of mathematics at Zurich Polytechnic, who considered him lazy (Einstein often skipped Minkowski’s classes).

Einstein thought Minkowski was too theoretical. They didn’t like each other much. Egos got in the way.

Reaction of Einstein and Laub to Minkowski’s ideas (Einstein starts to develop a more theoretical approach):

“In a recent published study Mr. Minkowski has presented the fundamental equations for the electromagnetic processes in moving bodies. In view of the fact that this study makes rather great demands on the reader in its mathematical aspects, we do not consider it superfluous to derive here these important equations in an elementary way, which, is, by the way, essentially in agreement with that of Minkowski”

Minkowski never made a priority claim and always gave Einstein his full share in the great discovery.

In 1915, six years after Minkowski’s death, his 1907 talk was published in the Annalen der Physik, and coordinated to the Annalen by Arnold Sommerfeld. Sommerfeld edited Minkowski’s text, and introduced a few changes to the original manuscript.

What is the legacy and what is our current understanding of these ideas?

Minkowski considered a world, space-time; he examined the transformation that leaves the expression:

c2t2 – x2 – y2 – z2 = 1

Invariant. This expression had already appeared in his two previous papers. He said that the structure of this equation with positive c consists of two hyperboloid sheets separated by t = 0. When transforming x, y, z, t into x’, y’, z’, t’, while y and z remain unchanged we draw the upper branch pf the hyperbola, the upper “light cone”: c2t2 – x2 = 1. And he presented new notions, “Weltpunkt x, y, z, t”, “Kerwe in der Welt”, “Weltlinie”, and “Weltlinien”.

Minkowski arrived at c2dt – dx2 – dy2 – dz2, which is positive, and thus the velocity v is always less than c.

Minkowski explained that according to Lorentz, any moving body, experiences a contraction in the direction of motion with a velocity v in the ratio 1:Ö(1 – v2/c2).

According to Einstein, the Lorentz contraction cannot be detected by physical means for an observer moving with the object. However, it can be detected by physical means by a non-co-moving observer. This is embodied in Minkowski’s graphical space-time “world”. (Weinstein 2012)

In 1900 experimental physics was physics. Resistance to relativity was partially caused by the maths not being available. It had to be developed.

The gravitational constant (also known as the “universal gravitational constant”, the “Newtonian constant of gravitation”, or the “Cavendish gravitational constant”), denoted by the letter G, is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton’s law of universal gravitation and in Albert Einstein’s general theory of relativity.

In Newton’s law, it is the proportionality constant connecting the gravitational force between two bodies with the product of their masses and the inverse square of their distance. In the Einstein field equations, it quantifies the relation between the geometry of spacetime and the energy–momentum tensor.

The measured value of the constant is known with some certainty to four significant digits. In SI units its value is approximately 6.674 × 10−11 N·kg–2·m2.

The modern notation of Newton’s law involving G was introduced in the 1890s by C. V. Boys. The first implicit measurement with an accuracy within about 1% is attributed to Henry Cavendish in a 1798 experiment.

The Cavendish experiment, performed in 1797–1798 by British scientist Henry Cavendish, was the first experiment to measure the force of gravity between masses in the laboratory and the first to yield accurate values for the gravitational constant. Because of the unit conventions then in use, the gravitational constant does not appear explicitly in Cavendish’s work. Instead, the result was originally expressed as the specific gravity of the Earth, or equivalently the mass of the Earth. His experiment gave the first accurate values for these geophysical constants.

The experiment was devised sometime before 1783 by geologist John Michell, who constructed a torsion balance apparatus for it. However, Michell died in 1793 without completing the work. After his death the apparatus passed to Francis John Hyde Wollaston and then to Henry Cavendish, who rebuilt the apparatus but kept close to Michell’s original plan. Cavendish then carried out a series of measurements with the equipment and reported his results in the Philosophical Transactions of the Royal Society in 1798.

John Michell (25 December 1724 – 21 April 1793) was an English natural philosopher and clergyman who provided pioneering insights in a wide range of scientific fields, including astronomy, geology, optics, and gravitation.

The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially-symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which makes exact solutions very difficult to find.

The Kerr metric is a generalization of the Schwarzschild metric, discovered by Karl Schwarzschild in 1915, which described the geometry of spacetime around an uncharged, spherically-symmetric, and non-rotating body. The corresponding solution for a charged, spherical, non-rotating body, the Reissner–Nordström metric, was discovered soon afterwards (1916–1918). However, the exact solution for an uncharged, rotating black-hole, the Kerr metric, remained unsolved until 1963, when it was discovered by Roy Kerr. The natural extension to a charged, rotating black-hole, the Kerr–Newman metric, was discovered shortly thereafter in 1965.

The Gödel metric is an exact solution of the Einstein field equations in which the stress–energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a nonzero cosmological constant. It is also known as the Gödel solution or Gödel universe.

This solution has many unusual properties—in particular, the existence of closed time-like curves that would allow time travel in a universe described by the solution. Its definition is somewhat artificial in that the value of the cosmological constant must be carefully chosen to match the density of the dust grains, but this space-time is an important pedagogical example.

The solution was found in 1949 by Kurt Gödel.

Model allows time travel (conditions are actually considered improbable so are not actually seen) and doesn’t expand. Closed time-line paths and self-consistent histories.

The speed of light links space and time

The speed of light is finite — The universe ≠ The visible universe

Most things have known unknowns and unknowable unknowns

Unfortunately cosmology also has unknowable unknowns

The universe is expanding. General relativity predicted this but Einstein didn’t.

Space-time is being stretched. Great clusters of galaxies show this but local groups etc. don’t.

The cosmic microwave background (CMB) is an almost-uniform background of radio waves that fill the universe. The CMB is, in effect, the leftover heat of the Big Bang itself – it was released when the universe became cool enough to become transparent to light and other electromagnetic radiation, 100,000 years after its birth. At this time, the universe was filled with a hot, ionized gas. This gas was almost completely uniform, but did have slight deviations – spots that were slightly (1 part in 100,000) more or less dense. The slight changes in the intensity of the CMB across the sky (deviations of only than 1 part in 100,000) give us a map of the early universe. The picture below such a map, measured by the Wilkinson Microwave Anisotropy Probe (WMAP), a space-based microwave telescope for studying the CMB. By studying this map, astrophysicists have learned an enormous amount about the evolution and composition of the universe.

Chaotic inflation produces local uniformity and global diversity in an infinite universe. The chaotic inflationary universe model gives a real reason to expect that the geography of the universe beyond our horizon differs from that in the visible part of the universe.

If a region inflates then it generally creates within itself, on minute length scales, the conditions for further inflation to occur from many of its parts. This process can continue into the infinite future, with inflated regions producing further subregions that inflate, and so on ad infinitum. The process has no end. It has been called the “eternal” or “self-reproducing” inflationary universe (Linde A., Sci. American, 1994, vol. 32). As yet, it is not known whether it need have a beginning.

Will the universe expand forever?

The key stages in evolution of a universe containing matter, radiation and vacuum energy

Einstein’s theories explain the acceleration of the expansion of the universe

The far, far future:

All stars die;

Black holes evaporate;

Black holes might not be endless depths after all, and may be it is not impossible to get out of it; some energy might be able to escape them. This surprising discovery was first predicted by Stephen Hawking in 1974.

The phenomenon is called Hawking radiation. Hawking radiation scatters a black hole’s mass into space and over time will actually do this until there is nothing left, basically killing the black hole.